Section 13.5 Going Deeper: Average Rate of Change
It turns out that the idea of slope is extremely important in both practical and theoretical mathematical thinking. Slopes can be used to represent important relationships between variables, and they one of the core concepts that we see in calculus. So we want to take extra time to really think about how we can understand and interpret the notion of slope.
Let's imagine that we're watching someone run the 100-meter dash, and that it takes them exactly 10 seconds to reach the finish line. It would make perfect sense for us to say that they ran 10 meters per second. But what does that really mean? Let's look at a graph that represents the situation.
The only things we know about the graph are that the runner started was at the starting line at \(t = 0\) and at the finish line (100 meters) at \(t = 10\text{.}\) The speed of 10 meters per second is the result of taking the ratio of distance and time:
If the original fraction has a familiar feel, that's because it's the formula for slope. It's nothing more than the rise over the run. We can see this more clearly if we connect the two points to create a line.
But there's a problem with this graph. It does not represent the actual position of the runner at every moment in time. Sprinters don't run at a constant speed. When they start they're moving at a slow speed, and then speed up as they go. So the real graph may look more like this:
So let's go back and think about what we mean when we say that they ran 10 meters per second. This actually represents the average speed over the course of the race. If we only knew the starting point and the ending point, then we can calculate this value as our best estimate of the speed, even though something different may have happened in between.
From this picture, we can try to break things down further. What was the average speed in the first 5 seconds and the last 5 seconds? Now instead of using just the endpoints, we're adding a point in the middle.
Notice that we now have two slopes. One slope for the first part and another slope for the second part. In other words, we now have an average speed for the first half of the race and the second half of the race. The smaller slope at the beginning and larger slope at the end matches with our intuition that the sprinter is moving faster at the end of the race than the beginning.
We can see that theses lines do a better job of capturing the information of the graph, but it's still not very good. We can still see some rather large deviations between the straight lines and the curve. But why stop at just two divisions? Why not make a new division every second?
Now the dashed line matches very closely with the curve. It's still not perfect, but it's getting a lot better. We now have the average speed for every second of the race.
This leads us to an interesting question: What happens if we continue this process? Instead of every second, what if we pushed this to every half second? Or every tenth of a second? Or every hundredth of a second? We can get average speeds for smaller and smaller time intervals. How far can we push this idea? And what is the end result?